Stable geometric properties of analytic and harmonic functions

Given any sense preserving harmonic mapping f = h + (g) over bar in the unit disk, we prove that for all vertical bar lambda vertical bar = 1 the functions f(lambda) = h + lambda(g) over bar are univalent (resp. close-to-convex, starlike, or convex) if and only if the analytic functions F-lambda = h + lambda g are univalent (resp. close-to-convex, starlike, or convex) for all such lambda. We also obtain certain necessary geometric conditions on h in order that the functions f(lambda) belong to the families mentioned above. In particular, we see that if f(lambda) are univalent for all lambda on the unit circle, then h is univalent.